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Okun on Leaky Buckets Okun started out with an approval of the pure transfer principle, but remarked that in reality “the money must be carried from the rich to the poor in a leaky bucket. Some of it will simply disappear in transit, so the poor will not receive all the money that is taken from the rich.” [Okun (1975, p. 91).] Okun (1975, p. 91) put the following question to an outside ethical observer: “I shall not try to measure the leak now, because I want you to decide how much leakage you would accept and still support the Tax-and Transfer Equalization Act.” Okun (1975, p. 94) states his own view: “Since I feel obliged to play the far- fetched games that I make up, I will report that I would stop at the leakage of 60 percent in this particular example.” Okun (1975, p. 94) states that the tolerated leakage is an increasing function of the income level of the transferors: “If the proposed tax were to be imposed only on the handful of wealthiest American families with annual incomes above $ 1 million, you might well support the equalization up to a much bigger leakage. In fact, some people would wish to take money away from the super-rich even if not one cent reached the poor.”

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The Transfer Principle with Transaction Costs Progressive transfers of incomes cause income distributions to become more equally distributed. Hence, income inequality measures decrease and aggregate welfare increases in response to progressive transfers. Suppose transfers incur transaction costs: What is the maximum leakage of transaction costs such that a transfer still “pays at the margin” [in terms of leaving the degree of income inequality or aggregate social welfare intact]? This is a generalization of the transfer principle for transaction costs.

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The Leaky-Bucket Paradox Leaky-Bucket Inconsistency: It is generally wrong to expect that some fraction of the transfer has to arrive at the transferee when the degree of income inequality should be maintained [Seidl (2001), Hoffmann (2001), Lambert and Lanza (2006)]. Rather there exists a unique benchmark, such that this conjecture holds only for transfers below this benchmark. For transfers above this benchmark the transferee has to receive a higher amount than the transfer. For transfers across this benchmark the “transferee” has even to lose some money to maintain the same degree of income inequality.

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The Universe of Pairwise Income Transactions Transfers may be regressive rather than progressive. Income earners may experience income increases rather than decreases. This yields a fourfold pattern of cases. Traditional research has focused only on the one case of progressive transfers.

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Theory The income inequality measure is an indicator of income inequality. For completely equal incomes it assumes the value zero. Increases in inequality are indicated by higher values of the income inequality measure. Thus, I(. ) is an increasing function of income inequality.

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The existence and the properties of benchmarks were first noticed by Seidl (2001) and Hoffmann (2001). A more comprehensive follow-up study is due to Lambert and Lanza (2006). Hoffmann (2001) uses the benchmark as some kind of poverty line to separate the “relatively poor” and the “relatively rich”. As the benchmark depends both on the income distribution and the inequality measure applied, this would make the poverty line dependent on the inequality measure applied.

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We now consider the basic result of the leaky bucket theory. It holds for differentiable, inequality averse and scale or translation invariant inequality measures. It will be experimentally tested. Case of progressive transfersSame conditions Change of inequality sign

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Nota bene: compensating justice [as observed in our experiments] is at variance both with the transfer principle and with leaky-bucket theory!

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Experimental Design Our experimental design consisted of the income distribution (500€, 750€, 1000€, 1250€, 1500€, 1750€, 2000€), where the incomes refer to monthly net incomes of seven equally numerous groups of income recipients. The income distribution was presented in the upper half of a computer screen. Upon touching a key, 100€ were either added or subtracted from one income at random. This was shown in the lower half of the computer screen (next slide). This gave us 7  6=42 combinations both for +100€ and -100€, that is, 84 combinations. Each subject responded to these 84 combinations (presented in a random order). There were no material incentives. The subjects were asked to adapt the second income in the lower half of the screen such that the degree of income inequality within this society should stay put.

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The Numerical Benchmarks By Theorem 10 the numerical benchmarks are implicitly defined by setting the partial derivatives of an income inequality measure equal to zero. The partial derivatives are functions of the income distribution and the parameters of the inequality measure. As the income distribution is given, the partial derivatives are functions of the inequality parameter alone. We use three income inequality measures: entropy, extended Gini, generalized Atkinson. Thus the benchmarks y* can be plotted for the domain of the inequality parameters. This is shown in Figure 2.

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Benchmark Functions for Note that the Atkinson inequality measure and the entropy inequality measure are just mirror images. An appropriate definition of the respective parameters can show that.

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Table 1 shows the parameter values for which the respective entries of the income distributions become the benchmarks. This table demonstrates that the parameter values for some entries at the bottom and at the top of the income distributions are completely implausible. Thus, Table 1 allows us to restrict the space in which y* may be located to the interval: 750< y*<1500.

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Compare what leaky-bucket theory asks for: (a) Negative responses  for  >0 and positive responses for  <0. Opposite in Table 2. (b) Positive responses  for  >0 and negative responses for  y k ). (c) For benchmarks symmetrically distributed around 1250€,  should be around zero. But Table 2 exhibits positive responses for  >0 and negative responses for  <0. Rather we observe compensating justice: Gains should entail gains, and losses should entail losses. Moreover, the focus of compensating justice rests on the richer party for income gains (the poorer party gets more) and on the poorer party for income losses (the richer party loses more). 

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Table 3 focuses on the numbers and percentages of responses. For income gains, positive income compensation dominates. For income losses, negative income compensation dominates. The focus of compensating justice rests on the richer party for income gains, and on the poorer party for income losses. If the poorer party experiences an income gain, richer persons receive income gains, too, but they are less than before (60.05% versus 76.98%). If the richer party experiences an income loss, poorer persons experience income losses, too, but they are less than before (52.50% versus 73.49%). Note, however, that the basic tendency remains intact.

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The second lines of the cells give the percentages of the same sides, opposite sides and unknown benchmarks. We do not observe diverse responses in the second lines of the cells of this table. Rather we observe a simple common pattern: income gains of one income recipient should be matched by income gains of the other involved income recipient to maintain inequality neutrality, and income losses should be matched by income losses of the other involved income recipient to maintain inequality neutrality. This is simple compensated justice. For δ>0 the poorer party should get more, for δ<0 the richer party should lose more. This is graded compensating justice. 76.98%

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Compensating Justice In order to capture these findings in quantitative terms, we estimated several equations. We finally settled on a logarithmic equation because all parameters with the exception of one are significant at the 10% significance level. Other equations used dummies for the relative position of the income recipient whose income was originally affected. But these equations, while they are more complicated and more difficult to interpret, did not provide a much better fit. Hence we estimated